One computer implemented approach for calculating a demand forecast involves defining a so-called demand forecast tree capable of being graphically represented by a single top level node with at least two branches directly emanating therefrom, each branch having at least one further node. The demand forecast is computed on the basis of historical sales data typically associated with bottom level nodes of a demand forecast tree by a forecast engine capable of determining a mathematical simulation model for a demand process. One such forecast engine employing statistical seasonal causal time series models of count data is commercially available from Demantra Ltd, Israel, under the name Demantra™ Demand Planner.
Demand forecast applications include determining the optimal draw matrix D* to maximize the expected total profit (ETP) realizable by a distribution policy for a single period inventory system given by:
                    ETP        =                                            ∑                              i                ⁢                                                                  ⁢                j                                      ⁢                          Ep              ⁡                              (                                  D                                      i                    ⁢                                                                                  ⁢                    j                                                  )                                              =                                    ∑                              i                ⁢                                                                  ⁢                j                                      ⁢                                                         [                                                                            (                                                                        p                                                      i                            ⁢                                                                                                                  ⁢                            j                                                                          -                                                  c                                                      i                            ⁢                                                                                                                  ⁢                            j                                                                                              )                                        ⁢                                          D                                              i                        ⁢                                                                                                  ⁢                        j                                                                              -                                                            (                                                                        p                                                      i                            ⁢                                                                                                                  ⁢                            j                                                                          -                                                  g                                                      i                            ⁢                                                                                                                  ⁢                            j                                                                                              )                                        ⁢                                          ER                      ⁡                                              (                                                                              λ                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                ,                                                      D                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                                      )                                                                              -                                                            b                                              i                        ⁢                                                                                                  ⁢                        j                                                              ⁢                                          EST                      ⁡                                              (                                                                              λ                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                ,                                                      D                                                          i                              ⁢                                                                                                                          ⁢                              j                                                                                                      )                                                                                            ]                                                                        Eqn        .                                  ⁢                  (          1          )                    where pij is the unit retail price of an ith consumer item at a jth location of the single period inventory system, cij is its unit production cost, gij is its unit return cost when unsold, and bij is its unit stockout cost. Derived from Eqn. (1), the optimal draw matrix D* for a single period inventory system is calculated using optimal availabilities Aij* where:
                              A                      i            ⁢                                                  ⁢            j                    *                =                              F            ⁡                          (                                                λ                                      i                    ⁢                                                                                  ⁢                    j                                                  ,                                  D                                      i                    ⁢                                                                                  ⁢                    j                                    *                                            )                                =                                                                      p                                      i                    ⁢                                                                                  ⁢                    j                                                  -                                  c                                      i                    ⁢                                                                                  ⁢                    j                                                  +                                  b                                      i                    ⁢                                                                                  ⁢                    j                                                                                                p                                      i                    ⁢                                                                                  ⁢                    j                                                  -                                  g                                      i                    ⁢                                                                                  ⁢                    j                                                  +                                  b                                      i                    ⁢                                                                                  ⁢                    j                                                                        .                                              Eqn        .                                  ⁢                  (          2          )                    In the case of the above-mentioned “newsvendor” and the “knapsack” problems, Eqn. (1) respectively degenerates to:
                    ETP        =                                            ∑              j                        ⁢                          EP              j                                =                                    ∑              j                        ⁢                          [                                                                    (                                                                  p                        j                                            -                                              c                        j                                                              )                                    ⁢                                      D                    j                                                  -                                                      (                                                                  p                        j                                            -                                              g                        j                                                              )                                    ⁢                                      ER                    ⁡                                          (                                                                        λ                          j                                                ,                                                  D                          j                                                                    )                                                                      -                                                      b                    j                                    ⁢                                      EST                    ⁡                                          (                                                                        λ                          j                                                ,                                                  D                          j                                                                    )                                                                                  ]                                                      and                          ETP        =                                            ∑              i                        ⁢                          EP              i                                =                                    ∑              i                        ⁢                                          [                                                                            (                                                                        p                          i                                                -                                                  c                          i                                                                    )                                        ⁢                                          D                      i                                                        -                                                            (                                                                        p                          i                                                -                                                  g                          i                                                                    )                                        ⁢                                          ER                      ⁡                                              (                                                                              λ                                                          i                              ⁢                                                                                                                                                                            ,                                                      D                                                                                                                                                      ⁢                              i                                                                                                      )                                                                              -                                                            b                      i                                        ⁢                                          EST                      ⁡                                              (                                                                              λ                            i                                                    ,                                                      D                            i                                                                          )                                                                                            ]                            .                                          
A distribution policy for a single period inventory system is often subject to one or more deterministic metric constraints, for example, a maximum total draw, a maximum budget, and the like, which necessitate an optimal constrained draw matrix denoted D^ whose total draw is typically less than the total draw of the optimal draw matrix D*. Two common approaches for solving such types of problems in the field of single-period inventory systems are the Lagrange multiplier approach as described in Silver, E., D. Pyke, and R. Peterson: Inventory Management and Production Planning and Scheduling, 3d ed., Wiley, NY, 1998, pgs. 406-422, and the one-by-one allocation or removal of draw units as discussed in Hadley, G., and T. M. Whitin, Analysis of Inventory Systems, Prentice-Hall, 1963, pgs. 304-307, the contents of which are incorporated herein by reference.